Howe duality and the quantum general linear group

Zhang, R.

doi:10.1090/S0002-9939-02-06892-2

Howe duality and the quantum general linear group
HTML articles powered by AMS MathViewer

by R. B. Zhang PDF

Proc. Amer. Math. Soc. 131 (2003), 2681-2692 Request permission

Abstract:

A Howe duality is established for a pair of quantized enveloping algebras of general linear algebras. It is also shown that this quantum Howe duality implies Jimbo’s duality between ${\mathrm U}_q({\mathfrak {gl}}_n)$ and the Hecke algebra.

References

Henning Haahr Andersen, Patrick Polo, and Ke Xin Wen, Representations of quantum algebras, Invent. Math. 104 (1991), no. 1, 1–59. MR 1094046, DOI 10.1007/BF01245066
Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
A. R. Gover and R. B. Zhang, Geometry of quantum homogeneous vector bundles and representation theory of quantum groups. I, Rev. Math. Phys. 11 (1999), no. 5, 533–552. MR 1696104, DOI 10.1142/S0129055X99000209
Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638
Michio Jimbo, Quantum $R$ matrix related to the generalized Toda system: an algebraic approach, Field theory, quantum gravity and strings (Meudon/Paris, 1984/1985) Lecture Notes in Phys., vol. 246, Springer, Berlin, 1986, pp. 335–361. MR 848629, DOI 10.1007/3-540-16452-9_{2}1
Bertram Kostant, On Macdonald’s $\eta$-function formula, the Laplacian and generalized exponents, Advances in Math. 20 (1976), no. 2, 179–212. MR 485661, DOI 10.1016/0001-8708(76)90186-9
A. N. Kirillov and N. Reshetikhin, $q$-Weyl group and a multiplicative formula for universal $R$-matrices, Comm. Math. Phys. 134 (1990), no. 2, 421–431. MR 1081014
Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316, DOI 10.1090/ulect/015
Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
Brian Parshall and Jian Pan Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), no. 439, vi+157. MR 1048073, DOI 10.1090/memo/0439
N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 193–225. MR 1015339
Mitsuhiro Takeuchi, Some topics on $\textrm {GL}_q(n)$, J. Algebra 147 (1992), no. 2, 379–410. MR 1161300, DOI 10.1016/0021-8693(92)90212-5